Title: Siegel's Lemma for polynomials irreducible over Q
Speaker: Michael Filaseta
Abstract: Siegel's Lemma is a statement about the finiteness of the number of integer points on an irreducible curve f(x,y) = 0 when the genus of the curve is at least 1. Here, we want f(x,y) to have rational coefficients and for irreducibility to be over the complex numbers (in C[x,y]). In this talk, we discuss how to restrict to the case where instead f(x,y) is irreducible over the rationals. The resulting statement is different but will give us what we want in forthcoming talks on deducing Hilbert's Irreducibility Theorem from Siegel's Lemma. The idea is to make this connection without much in the way of machinery, so the talk is intended for graduate students and faculty alike.
Title: Universal Hilbert sets
Speaker: Michael Filaseta
Abstract: A universal Hilbert set is an infinite set S of integers with the property that for every f(x,y) with integer coefficients with f(x,y) irreducible in Q[x,y] and of degree at least 1 in x, we have that for all but finitely many t in S, the polynomial f(x,t) is irreducible in Q[x]. We describe a set S with this property that has asymptotic density 1 in the integers. Then we show an important connection that S has with Siegel's Lemma.
Title: Universal Hilbert sets, Part II
Speaker: Michael Filaseta
Abstract: After reviewing a little from last week, we finish a proof of a lemma associated with Hilbert's Irreducibility Theorem and Siegel's Lemma. We also prove an additional result that encapsulates the use of Siegel's Lemma in proving Hilbert's Irreducibility Theorem. After this result, there will be no need to recall the statement of Siegel's Lemma or the definition of our example of a universal Hilbert set, as this result indirectly embeds all the information we will need.
Title: What is the height of two points in the plane?
Speaker: Frank Thorne
Abstract: A classical question in arithmetic geometry is to count the number of rational points of bounded height on algebraic varieties.
After reviewing some classical cases, I'll discuss this problem for the Hilbert scheme of two points in the plane. This quickly turns into a lattice point counting problem, and invites related questions as well as arithmetic applications.
I'll spend a lot of the talk explaining what all the words mean.
Joint work with Jesse Kass.
There is no seminar this week; instead people are encouraged to attend talks by Michael Burr in the Algebra/Algebraic Geometry Seminar starting at 2:30 pm in LC 303B.
There is no seminar this week, but you can see Number Theory talks at the SERMON meeting listed below.
Title: Universal Hilbert sets: the final details
Speaker: Michael Filaseta
Abstract: After a quick review of what has been done thus far, we will show how to establish that a certain explicit set S is a universal Hilbert set with asymptotic density 1 in the integers.
Title: What is the height of two points in the plane?, Part II
Speaker: Frank Thorne
Abstract (Part I version): A classical question in arithmetic geometry is to count the number of rational points of bounded height on algebraic varieties.
After reviewing some classical cases, I'll discuss this problem for the Hilbert scheme of two points in the plane. This quickly turns into a lattice point counting problem, and invites related questions as well as arithmetic applications.
I'll spend a lot of the talk explaining what all the words mean.
Joint work with Jesse Kass.
There is no seminar this week.
Title: Divisor-sum fibers
Speaker: Lola Thompson (Oberlin College)
Abstract: Let s( ) denote the sum-of-proper-divisors function, that is, s(n) = sum of d as d ranges over the divisors of n. Erdos-Granville-Pomerance-Spiro conjectured that, for any set A of asymptotic density zero, the preimage set s^{-1}(A) also has density zero. We prove a weak form of this conjecture. In particular, we show that the EGPS conjecture holds for infinite sets with counting function O(x^{(1/2) + epsilon(x)}). We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers alpha and epsilon, there are integers n with arbitrarily many s-preimages lying between alpha(1-\epsilon)n and alpha(1+epsilon)n. This talk is based on joint work with Paul Pollack and Carl Pomerance.
Title: Densities of sets of integers represented by quadratic forms
Speaker: Pete Clark (University of Georgia)
Abstract: Given an integral quadratic form, one seeks to determine the set of integers that it represents. Though classical -- sums of squares were handled by Fermat, Lagrange and Legendre-Gauss -- in general the problem remains open, notwithstanding the algebraic, arithmetic and analytic theories developed to address it. In this talk we investigate the size of the set of integers represented. The main result is a "Near Hasse Principle" -- an integral quadratic form represents 100% of the integers that it locally represents. This leads to an asymptotic for the number of integers of absolute value at most X represented. When there are at least three variables, this amounts to determining the density of the set of represented integers. We then address -- but do not fully resolve -- the inverse problem of which densities arise. This is joint work with Paul Pollack, Jeremy Rouse and Kate Thompson.
Title: A quest for positive definite matrices in finite fields (a work in progress)
Speaker: Erin Hanna
Abstract: Positive definite matrices make up an interesting and extremely useful subset of Hermitian matrices. These matrices admit a plethora of equivalent statements and properties, one of which is an existence of a unique Cholesky decomposition. We consider whether any of these equivalent statements to having a unique Cholesky can be analogized for matrices over finite fields. We present new definitions for positive definite matrices over finite fields and some equivalences seen to still hold. Currently, results have been shown for finite fields of certain sizes, specifically powers of two and odd powers of primes congruent to 3 mod 4. There will be room for suggestions on how to proceed with the other finite fields and continuing results in the current cases.
Title: TBA
Speaker: TBA
Abstract: TBA